Which version of polymake are you using? We just released version 3.3, and there it works.

It is possible that in some older versions the types get messed up. Beware that the property POINTS takes a small object of type Matrix<TropicalNumber<Min>>. Object of type Rational should automatically convert to TropicalNumber<Min> but not conversely. Maybe you can solve your problem (even in an old version of polymake) by explicitly creating the matrix object first and then use it to construct the tropical polytope.

$m = new Matrix<TropicalNumber<Min>>([[0,1/2,4]]);
polymake: ERROR: Shared module compilation failed; most likely due to a type mismatch. Set the variable $Polymake::User::Verbose::cpp to a positive value and repeat for more details.

I have version 3.2 of polymake and I'm running a Mac so I guess I need to wait a bit for the release of 3.3 for Mac.